Real time asymptotic packing

A random greedy algorithm, somewhat modified, is analyzed by using a real time context and showing that the variables remain close to the solution of a natural differential equation. Given a (k + 1)-uniform simple hypergraph on N vertices, regular of degree D, the algorithm gives a packing of disjoint hyperedges containing all but O(ND−1/k lnD) of the vertices. Let H = (V, E) be a (k+ 1)-uniform hypergraph on N vertices. A packing P is a family of disjoint edges. Given P we correspond the set S = V − ⋃ P of those vertices v not in the packing, these v we call surviving vertices. We shall assume: • H is simple. That is, any two vertices are in at most one edge. • H is regular of degree D. That is, every vertex v lies in precisely D e ∈ E. We are interested in the asymptotics for k fixed, D,N → ∞. We assume k ≥ 2 is fixed throughout. We show Theorem. There exists a packing with |S| = O(ND lnD) where c depends on k. (We make no attempt to optimize c.) Our approach is to give a real time random process that produces a packing with E[|S|] meeting these bounds. The process, as described in §1,2, can be thought of as the random greedy algorithm with some “stabilization mechanisms” added. Placing the algorithm in a real time context allows for simulation of the variables by a differential equation and the analysis of our discrete, albeit asymptotic, procedure becomes quite continuous in nature. The study of asymptotic packing can be said to date from the proof by V. Rödl [3] of a classic conjecture of Paul Erdős and Haim Hanani [2]. Rödl showed that for l < k fixed and n→∞ there exists a “packing” P of ∼ ( n l ) / ( k l ) k-element subsets of an n-element universe Ω so that every l points of Ω lie in at most one of the k-sets. This was nicely generalized by N. Pippenger in work appearing [5] jointly with this author. He showed that any k-uniform hypergraph on N vertices with deg(v) ∼ D for every v and any two vertices v,w having o(D) common edges has a packing P with |S| = o(n). (Here k is fixed, N,D →∞.) Recent work has centered on lowering the size of |S| in terms of D. Our main result has also been shown (indeed, without the logarithmic term for k ≥ 3) in our joint paper [1] by quite different techniques. AMS(1991) Subject Classification: Primary 05B40, Secondary 60D05 the electronic journal of combinatorics 4 (no. 2) (1997), #R19 2 1 Two Simple Algorithms We first define the discrete random greedy algorithm in a natural way. Randomly order e1, . . . , eω, ω = |E|, the edges of H. Set P0 = ∅, S0 = V . For 1 ≤ i ≤ ω if ei ⊆ Si−1 then set Pi = Pi−1 ∪ {ei} and Si = Si−1 − ei, else keep Pi = Pi−1 and Si = Si−1. That is, consider the edges in random sequential order and add each to the packing if you can. We conjecture that E[|Sω|] meets the bounds of our Theorem. This author [6] and, independently, V. Rödl and L. Thoma [4] have shown that E[|Pω|] ∼ N k+1 or equivalently that E[|Sω|] = o(N). Viewed in this light we are now looking at a second order term, just how close to a “perfect packing” can we get. Unfortunately, this natural algorithm has eluded more refined analysis. We feel it would be most interesting even to prove that the exponent of D is the correct one, that E[|Sω|] = O(ND ) (??) (1) Now we define the realtime random greedy algorithm. We let time t go continuously starting from zero. The packing P = Pt will vary with time as will St = V − ⋃ Pt. We let Ht denote the restriction of H to St. If by time t edge e ⊆ St has not yet been born then it is born in the next dt with probability e dt kD . When e is born it is added to P . In particular, all e with e ∩ e 6= ∅ are no longer considered. Observe that the edges are being born in a random order. Thus if we continue this process until H has no edges the distribution of S will be precisely that of the discrete random greedy algorithm. It will be more convenient, however, to stop the process at time ω = lnD. We now give a heuristic guide which should motivate the full process we define later. Let degt(v) be (for v ∈ St) the degree of v in Ht and suppose all degt(v) ∼ f(t)D. There would be ∼ kf (t)D pairs (e, e) where e is an edge containing v and e is an edge intersecting e, but not at v. Each e is born in the next dt with probability e dt kD and if born diminishes deg(v) by one for each (e, e). (If e itself is born then v is removed from H.) On average deg(v) is decreased by kf(t)De dt kD = ef(t)D · dt. If this is to be f(t+ dt)D then we would need f(t+ dt) = f(t)− ef(t)dt f (t) = −ef(t) so that, as f(0) = 1, we would have f(t) = e. Indeed, the choice of birth intensity was designed so that f (t) would have this particularly convenient form. Suppose v has survived to time t. It lies on ∼ De edges, each is born with probability e dt Dk in the next dt so v is removed from S with probability dt k . The probability that v survives to time t starting at time zero would then be exp[− ∫ t 0 dt k ] = e. Since we want deg(v) ∼ De but deg(v) is integral we can only hope to carry this approximation through time ω = lnD. At that time Pr[v ∈ St] would be e = D. By Linearity of Expectation we would have E[|Sω|] = ND. the electronic journal of combinatorics 4 (no. 2) (1997), #R19 3 As we said earlier we are unable to make this argument rigorous and it is only conjecture that the result is correct. We see the basic problem as one of stability of a random system. The values degt(v) are random variables that will naturally oscillate around their means. The difficulty is that once some degt(v ) are abnormally off their mean then it affects the change in degt(v). (If v , v have a common edge e then deg(v) affects the number of (e, e) which affects the expected change of deg(v).) The N different degt(v) are all oscillating off their means and the oscillation of one can have an adverse affect on the oscillations of another. To handle this problem we modify the realtime random greedy algorithm by what we think of as stabilization mechanisms.